NeuralEstimators

NeuralEstimators facilitates a suite of neural methods for parameter inference in scenarios where simulation from the model is feasible. These methods are likelihood-free and amortised, in the sense that, once the neural networks are trained on simulated data, they enable rapid inference across arbitrarily many observed data sets in a fraction of the time required by conventional approaches.

The package supports neural Bayes estimators (NBEs), which transform data into point summaries of the posterior distribution; neural posterior estimators (NPEs), which perform approximate posterior inference via KL-divergence minimisation; and neural ratio estimators (NREs), which approximate the likelihood-to-evidence ratio and thereby enable frequentist or Bayesian inference through various downstream algorithms.

User-friendliness is a central focus of the package, which is designed to minimise "boilerplate" code while preserving complete flexibility in the neural-network architecture and other workflow components. The package accommodates any model for which simulation is feasible by allowing users to define their model implicitly through simulated data. A convenient interface for R users is available on CRAN.

Installation

To install the package, please first install the current stable release of Julia. Then, one may install the current stable version of the package using the following command inside Julia:

using Pkg; Pkg.add("NeuralEstimators")

Alternatively, one may install the current development version using the command:

using Pkg; Pkg.add(url = "https://github.com/msainsburydale/NeuralEstimators.jl")

Quick start

In the following minimal example, we develop a neural estimator for   from data  , where each   and we use the marginal priors   and  .

Point estimation

using NeuralEstimators, Flux, CairoMakie

# Functions to sample from the prior and simulate data
d, m = 2, 100  # dimension of θ and number of replicates
sampler(K) = NamedMatrix(μ = randn(K), σ = rand(K))
simulator(θ::AbstractVector) = θ["μ"] .+ θ["σ"] .* sort(randn(m))
simulator(θ::AbstractMatrix) = reduce(hcat, map(simulator, eachcol(θ)))

# Neural network, an MLP mapping m inputs into d outputs
network = Chain(Dense(m, 64, gelu), Dense(64, 64, gelu), Dense(64, d))

# Initialise a neural estimator
estimator = PointEstimator(network)

# Train the estimator
estimator = train(estimator, sampler, simulator)

# Assess the estimator
θ_test = sampler(250)
Z_test = simulator(θ_test)
assessment = assess(estimator, θ_test, Z_test)
bias(assessment)
rmse(assessment)
plot(assessment)

# Apply to observed data (here, simulated as a stand-in)
θ = sampler(1)                   # ground truth (not known in practice)
Z = simulator(θ)                 # stand-in for real observations
estimate(estimator, Z)           # point estimate

Posterior estimation

using NeuralEstimators, Flux, CairoMakie

# Functions to sample from the prior and simulate data
d, m = 2, 100  # dimension of θ and number of replicates
sampler(K) = NamedMatrix(μ = randn(K), σ = rand(K))
simulator(θ::AbstractVector) = θ["μ"] .+ θ["σ"] .* sort(randn(m))
simulator(θ::AbstractMatrix) = reduce(hcat, map(simulator, eachcol(θ)))

# Neural network, an MLP mapping m inputs into d outputs
network = Chain(Dense(m, 64, gelu), Dense(64, 64, gelu), Dense(64, d))

# Initialise a neural estimator
estimator = PosteriorEstimator(network, d; num_summaries = d)

# Train the estimator
estimator = train(estimator, sampler, simulator)

# Assess the estimator
θ_test = sampler(250)
Z_test = simulator(θ_test)
assessment = assess(estimator, θ_test, Z_test)
bias(assessment)
rmse(assessment)
plot(assessment)

# Apply to observed data (here, simulated as a stand-in)
θ = sampler(1)                   # ground truth (not known in practice)
Z = simulator(θ)                 # stand-in for real observations
sampleposterior(estimator, Z)    # approximate posterior sample

Ratio estimation

using NeuralEstimators, Flux, CairoMakie

# Functions to sample from the prior and simulate data
d, m = 2, 100  # dimension of θ and number of replicates
sampler(K) = NamedMatrix(μ = randn(K), σ = rand(K))
simulator(θ::AbstractVector) = θ["μ"] .+ θ["σ"] .* sort(randn(m))
simulator(θ::AbstractMatrix) = reduce(hcat, map(simulator, eachcol(θ)))

# Neural network, an MLP mapping m inputs into d outputs
network = Chain(Dense(m, 64, gelu), Dense(64, 64, gelu), Dense(64, d))

# Initialise a neural estimator
estimator = RatioEstimator(network, d; num_summaries = d)

# Train the estimator
estimator = train(estimator, sampler, simulator)

# Assess the estimator
θ_test = sampler(250)
Z_test = simulator(θ_test)
assessment = assess(estimator, θ_test, Z_test)
bias(assessment)
rmse(assessment)
plot(assessment)

# Apply to observed data (here, simulated as a stand-in)
θ = sampler(1)                   # ground truth (not known in practice)
Z = simulator(θ)                 # stand-in for real observations
sampleposterior(estimator, Z)    # approximate posterior sample

Contributing

To get started, see CONTRIBUTING.md for an overview of the code structure, development workflow, and how to submit contributions. A list of planned improvements is available in TODO.md, and instructions for contributing to the documentation can be found in docs/README.md.

If you encounter a bug or have a suggestion, please feel free to open an issue or submit a pull request.

Supporting and citing

This software was developed as part of academic research. If you would like to support it, please star the repository. If you use it in your research or other activities, please also use the following citation.

@misc{NeuralEstimators.jl,
  title = {{NeuralEstimators.jl}: A {J}ulia package for efficient simulation-based inference using neural networks},
  author = {Sainsbury-Dale, Matthew},
  year = {2026},
  note = {Version 0.2.0},
  howpublished = {\url{https://github.com/msainsburydale/NeuralEstimators.jl}}
}

Papers using NeuralEstimators

• Likelihood-free parameter estimation with neural Bayes estimators [paper] [code]

• Neural methods for amortized inference [paper][code]

• Neural Bayes estimators for irregular spatial data using graph neural networks [paper][code]

• Neural Bayes estimators for censored inference with peaks-over-threshold models [paper] [code]

• Neural parameter estimation with incomplete data [paper][code]

• Neural Bayes inference for complex bivariate extremal dependence models [paper][code]

• Fast likelihood-free parameter estimation for Lévy processes [paper]